DNA polymorphisms such as SNP and familial effects (additive genetic, common environment) to

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1 SUPPLEMENTARY MATERIALS, B. BIVARIATE PEDIGREE-BASED ASSOCIATION ANALYSIS Introduction We propose here a statistical method of bivariate genetic analysis, designed to evaluate contribution of the DNA polymorphisms such as SNP and familial effects (additive genetic, common environment) to variation of two interrelated traits without predefined distribution (i.e. could be quantitative and/or qualitative phenotypes). Our approach is an alternative of the liability-threshold concept (Falconer, 1), and it is based on the discrete models of genetic and familial effects. It can be helpful in testing the questions such as: Whether the observed two-dimensional phenotypic distribution may be explained without presuming common additive gene effects? What are the relative contributions of the specific and common environmental and genetic factors to variation of each of the two traits? Let and Y are two traits, where is a quantitative trait and Y is a binary trait (until further notice). Let G be a bi-allelic SNP (genotypes: aa, ab, bb) associated with both traits. In order to take into account additive effect of the other genes on the traits', we introduce three independent binary factors Z, Z Y, and Z Y (Fig. A1). In our model they represent genetic factors affecting variation of each trait separately (Z and Z Y ) and both traits simultaneously (Z Y ), pleiotropic effect. It is obvious that a binary factor can t express the entire variation caused by additive genetic effect. However, the specific portion of the variation explained by binary variable can be assigned. Gene-independent effects, caused by random or common familial effects on the phenotype variation are also taken into account in the model as provided below. 1

2 Fig. A1. Outlines of the model presenting factors that are taking into account in the analysis of two traits; and Y. Non-genetic factors are marked with rectangle. Model formalization and simplifications Binary model of additive genetic effect on trait variation The genetic component, in the proposed regression model (see below), is represented by random binary variables Z: Z=1 or Z=-1. The probability of a child value (Z C ) depends on the parental values, Z M and Z F, according to formulae presented in Table A1. Table A1. Probability of a child Z value conditioned by the parental values, P(Z C /Z F,Z M ). Z F Z M Z C P(Z C /Z F,Z M ) p* p +1 p + Δ -1 (1 - p) - Δ +1 p - Δ -1 (1 - p) + Δ *p = q + Δ(1- q), 0< q<1, Δ<1/. Where q is the frequency of positive genetic effect (Z=+1),

3 Δ is the extent of the heritability (transmittable factors) effect. This two-parametric model has the number of properties. The first parameter q is stationary, i.e. the expected value of q remains constant from generation to generation, like the allele frequency in the classical equilibrium model (see downmost Lemma 1). Correspondingly, the expected population s variance of Z doesn t depend on the second parameter Δ. The expected portion of Z C variance attributable to Z M and Z F values is Δ (see Lemma ). In case Δ=0. the correlation between the relatives (Table A) corresponds to the classical model of additive genetic effects (Falconer and Mackay, 1). Table A. Intra-family Z value correlations* Relatives with known Z value Explained variance Explained variance (h =0.) One of the parents 0. h 0. Both of the parents h 0.0 One of the sibs h 0. One of the monozygotic twins *h =Δ The proportion of the phenotypic variation of the study trait attributable to Z components of the model are estimated implementing the below presented bivariate regression model. Bivariate regression model In addition to SNP genotype (G), the discrete components of the regression model include three genetic components (Z & Z Y & Z Y), defined above. They could be interpreted in terms of the additive genetic and three common family environment components (F & F Y & F Y ). The Z, F and Z Y, F Y represent phenotype specific genetic and environmental factors, Z Y and F Y - represent genetic an environmental factors shared by both phenotypes. The regression analysis performed under the following simplifications (S1-S): S1. Family distribution of Z values (Z & Z Y & Z Y) is defined by the - binary model (see above) under Δ=0., while monozygotic twins shared the same Z values. The common family environment factors are

4 represented by binary variables (F & F Y & F Y). The F values are equally probable, i.e. P(F=+1) = P(F=- 1) = 0., and the offspring of the same parents share the same F values. Population's allele-frequency of the SNP is known, the genotype frequencies and the transmission probabilities follow the Hardy- Weinberg and Mendelian expectations, respectively. S. The trait has a normal distribution, P()=N[M, ]; where the average, M, depends on the individual genetic (G, Z and Z Y) and familial (F and F Y) factors, is the trait variation independent on the above factors. The sample average and the standard deviation are m = 0 and σ =1, respectively. Effect of G, Z, Z Y, F and F Y on the -average is additive and defined by linear regression model: M = m + G A α 1 + Z α + Z Y α + F α + F Y α G ( g p)/ p(1 p), where g is the SNP genotype (namely, the minor allele dose: 0, 1 or ), p is A the minor allele frequency; α 1, α, α, α and α unknown parameters. S. The Y trait has a binomial distribution P(Y=µ) = (M Y ) µ (1- M Y ) 1-µ ; where 0<M Y <1, µ=0 or µ=1. Effect of individual genetic (G, Z Y, Z Y) and family factors (F Y, F Y) on Y average (M Y), is additive and defined by logit model: ln(p(y=1)/p(y=0)) ln(m y/(1-m y)) +G Z Z F F Y1 Y Y Y Y Y Y Y Y Where m Y is the sample frequency of individuals with Y=1; β Y1, β Y, β Y, β Y, β Y are unknown parameters. Similar model of binary trait variation was previously described (Mukhopadhyay et al, 0). S. The factor-independent correlation between and Y was modeled with the following equations: P( Y, ) PY ( ) P ( / Y); P ( / Y) NM [ '( ), ' ]; ' 1 r M'( / Y1) M( ) r PY ( 1) / PY ( 1) ; M'( / Y1) M( ) r PY ( 1) / PY ( 1) ;

5 Where M() = M and P(Y=1) = M Y are defined above (the equations S1-S); r is the correlation parameter (-1<r< 1). S. In case of two quantitative traits, the two-dimensional normal distribution was applied: P(,Y)=N[M(),M(Y),, Y,r]. Where M(Y) and Y are defined by the same way as described in S1-S above. In case of two binary traits, the two-dimensional binomial distribution was applied: PY (, ) P ( ) PY ( ) sr Y. Where P() is defined by the same way as described in S above; s sign[( P ) (Y PY )] ; Y and Y are standard deviation of the traits: (1 P( 1)) P( 1), Y (1 PY ( 1)) PY ( 1). In case of mixture of traits (quantitative and binary) the model includes twelve parameters: α 1, α, α, α, α,, β Y1, β Y, β Y, β Y, β Y and r. In case of two quantitative or two binary traits the model includes 1 and parameters, respectively. Variance component estimation Since in our discrete factors model the genetic variation is approximated by using binary factors (predictors), the genetic effect on the trait s variation can be underestimated, while the environmental effect - overestimated accordingly. This source of systematic bias could be taking into account and corrected. Let Z N, F N and effects, respectively; parameters S N are unknown additive genetic, common family and individual environment Z B, F B and NormalDistribution.html). It defines the first equation: S B are the variance component estimates (the squares of the regression coefficients) obtained using binary approximation of additive genetic component. In case of a sample of twins the following three equations define the relations between the first and the second sets of values. Assuming the normal distribution of the additive genetic component, the proportion of its variation explained by the binary approximation is /π (

6 Z B Z N Remaining part of the additive genetic variance, 1-/π, is the mixture in F B and in variance estimates. S B On average, additive genetic variance shared by sibs equals to (1+P MZ)/; where P MZ is frequency of monozygotic twins among sib-pairs. The second equation is F (1 ) 1 P MZ / B F N Z N The residual, i.e. equation is 1 1+P / 1P /, is the mixture in MZ MZ S (1 ) 1 P MZ / B S N Z N S B variance estimate. Then the third Thus there is reciprocation between variance estimates obtained in our binary approximation model and the Z, N F N and S N values, i.e. ; ; Z N Z B FN FB 1 PMZ SN S 1 PMZ. B The last equations should be applied to both common and trait specific variance component estimates obtained using binary model of additive genetic effect. Limitation on input data The present version of the software has a limited option for simultaneous testing and adjustment for covariates and reserved for examination of the SNP effect. Currently, adjustment for other covariates could be done prior to analysis, taking into account familial structure of the sample. There are several available software, such as MAN (Malkin and Ginsburg, 01) that could manage this procedure. Adjustment of quantitative traits is usually achieved by using polynomial functions, while the qualitative phenotypes could be adjusted using logistic regression and/or the Gompertz-Makeham functions (e.g.

7 Korostishevsky et al. 01). Correspondent regression functions could replace m and/or m Y in formulae S and S. Likelihood evaluation and statistical inferences Consider first unrelated individuals. In case of n unrelated individuals the model-based likelihood (LH) of the observation ({ & Y & G}, i 1,,, n) is calculated by formula: i i i LH P( Z ) P( F) ( / G & Z & F & Y) P( Y / G & Z & F) i Z, F i i i i i i i i i i i 1 1 Where Z and F are three- and one-dimensional variables, respectively; ( / G & Z & F & Y) is the density of the normal distribution: N[ M( i / Gi & Zi & Fi & Yi), ]. i i i i i In case of pedigree data, the LH value also depends on correlations between relatives caused by genetic and environmental factors. In our model these correlations are defined by the following three discrete functions: P(G C/G F,G M), P(Z C/Z F,Z M) and P(F C/F SIB). The LH computation on pedigree data was previously repeatedly described (Ott J, 1; Lange K, Elston RC. 1). Comparison of the general and restricted models is based on the likelihood ratio test (LRT). SD of the maximum likelihood estimates (MLE) is evaluated following the asymptotic approximation: ( Lemma 1. * * / LRT The binary model of genetic variation (see Table A1 for definitions) is stationary by the q parameter. Proof: Let q be the frequency of individuals possessing Z=1 in the parent generation. Then, the expected frequency of individuals possessing Z=1 in the offspring generation (Q), satisfies the following equation: Q= q (1 - q) p+ q (p + Δ) + (1 - q) (p Δ) Where p = q + Δ(1- q). The direct substitution leads to the following equation: Q = q

8 Lemma. The expected portion of the Z variance attributable to parents, h, equals to Δ. Proof: The population variance of a binary trait Z (e.g. Z=1 or Z=0) is defined in the ordinary way: Var pop = q (1 - q) For known parent combinations of Z values (01, or 00), the child variance of Z value is defined according the binary model of genetic variation (Table A1): Var 01 = p(1 - p); Var = (p + Δ)(1 - p - Δ); Var 00 = (p + Δ)(1 - p - Δ) For random family, the expected child variance is: Var exp = q(1-q)var 01+q Var + (1 - q) Var 00 = q (1-q) (1- Δ ) The direct substitution leads to Var exp = q (1-q) (1- Δ ) Accordingly to this, the portion of Z variance explained by the parents values is h = = 1 Var exp /Var pop Δ In the same way, one can find out the portions of Z variance explained by one of the parents values, by the sib s value and by the twin s value (see Table A). In this context, h for generally used quasi genetic quantitative trait - number of specific SNP allele copies (0, 1, ), is equal to 0.. 1

9 1 1 1 References Falconer DS. Introduction to Quantitative Genetics, rd ed. UK/New York: Longmans Green/John Wiley & Sons, Harlow, Essex; 1. Korostishevsky M, Williams F, Hart D, Blumenfeld O, Spector T, Livshits G. Implementation of the simplified stochastic model of ageing for longitudinal osteoarthritis data assessment. Ann Hum Biol. 01;():1-. Lange K, Elston RC. Extensions to pedigree analysis I. Likehood calculations for simple and complex pedigrees. Hum Hered. 1;():. Mukhopadhyay I, Saha S, Ghosh S. Integrating binary traits with quantitative phenotypes for association mapping of multivariate phenotypes. BMC Proceedings, 0;.Suppl :S Ott J. Estimation of the recombination fraction in human pedigrees: efficient computation of the likelihood for human linkage studies. Am J Hum Genet. 1;():. Weisstein, Eric W. "Half-Normal Distribution." From MathWorld--A Wolfram Web Resource. Half-NormalDistribution.html